For a solenoid...why did the secondary voltage change when the iron core was pulled out of the inner coil?

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If you disrupt the setup of the primary coil with its set of primary windings in relation to the secondary coil with its secondary windings, you will cancel the effect of the induced electromagnetic force.  The primary coil has a current passed through it, which induces a magnetic field in the ferromagnetic core the wire is wrapped around.  This magnetic field in turn induces an electric current through the coils of the secondary winding.  This current may be stepped-up by having twice as many coils or stepped-down by having half as many coils.  This important concept has not changed in design much since Michael
Farraday made the discovery that passing an electric current through a wire will produce an electromagnetic field that can influence other materials that will conduct electricity and magnetism.  So if you upset the arrangement of the two coils, it will otherwise cancel the desired electrical effect.

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